Relativistic effects cause error in time reading (v2):

RG DOI : https://doi.org/10.32388/3YQQBO.2 

Abstract

The paper, titled 'Relativistic effects cause errors in time reading', highlights how the concept of time dilation, a consequence of the theory of relativity, creates different time scales for proper time and time dilation. This difference in time scale introduces errors in clock readings when attempting to measure time dilation using the same units as proper time.

The theory of relativity adopts Minkowski spacetime which combines three-dimensional Euclidean space and fourth-dimensional time into a four-dimensional manifold, where time is stripped of its independence, rather considered 'natural'. The theory of relativity also implies that proper time (t) is dependent on relativistic effects and is expressed as 𝑡 < 𝑡′, where t' is the time dilation. The equation for time dilation is 𝑡՚ = 𝑡/√(1 − 𝑣²/𝑐²) where 𝑡′ is dilated time, 𝑡 is proper time, v is relative speed and c is the speed of light in free space.

Experiments carried out in the electronics laboratory on piezoelectric crystal oscillators show that the waves correspond to changes in time due to relativistic effects. where the time interval 𝑇(𝑑𝑒𝑔) is inversely proportional to the frequency (𝑓), for 1° phase. We get a wave associated with time change. For example, a 1° phase shift in a 5 MHz wave corresponds to a time change of 555 picoseconds (ps). Phase shifts in relative frequency, due to motion or gravitational potential differences, correspond to wavelength enlargement of clock oscillations in the clock mechanism, resulting in errors in clock readings. [1]

As per the Special Theory of Relaitivity, time dilation results from relativistic effects, such as speed or gravitational potential difference, that cause time to run differently for the moving object compared to an observer at rest.[2] Due to this difference, the time dilation (𝑡՚) cannot be directly measured using the same time scale (clock) used to measure proper time (𝑡).

Mathematical Representation:

We know, the equation of time dilation due to speed is 𝑡՚ = 𝑡/√(1 − 𝑣²/𝑐²); where, 𝑡՚> 𝑡;  

The instantaneous phase (ϕ) represents an angular shift between two relative sine waves and is measured in degrees. After a time Δt, the two relative sine waves are initially synchronized in phase but differ in frequency by Δω degrees per second, developing a differential total phase shift (ΔΦ). Eq. Given by: ΔΦ = Δω × Δt.

The time interval 𝑇(𝑑𝑒𝑔) is inversely proportional to the frequency (𝑓);

Where the time shift ∆t, due to the speed or gravitational potential difference, represents the error in the exact time (t) and consequently t < t'; For mathematical and geometric reasons as described below.

𝑇(𝑑𝑒𝑔) = 𝑇/360 = (1/𝑓)/360 = ∆t; Time scale = 360 (𝑇/360); t < t';

Time scale for Proper time = 360°; Proper time = t; 

Time scale for Time dilation > 360°; Time dilation = t';  

Since, [Time scale for Proper time]  ≠  [Time scale for Time dilation];

Therefore, Time scale (clock) for Proper time cannot display Time dilation.

The time scale for proper time (t) and the time scale for time dilation (t') are different. The time scale for proper time (t) is 360°, as represented by the 𝑇(𝑑𝑒𝑔) = 𝑇/360 equation. The time scale for time dilation (t') is greater than 360°, Since, Time scale for Time dilation > 360°, and the two time scales are not the same, the clock that measures proper time (t) cannot display or measure time dilation (t') in the same units.

Conclusion: 

Propoer time (t) and time dilation (t') are associated with different time scales, and a clock that measures proper time cannot directly display or measure time dilation in the same unit. The relativistic effect of time dilation causes time to dilate or stretched for a moving object or object in gravitational potential difference relative to an observer at rest, which creates different time scales for proper time and time dilation, where, due to motion or gravitational potential, the phase changes in relative frequency corresponds to an increase in the wavelength of the clock's oscillation, which results in an error in the clock's reading.

In short, proper time and time dilation have different time scales, causing errors in clock time reading.

Reference:

[1]  Thakur, Soumendra Nath. Effect of Wavelength Dilation in Time.-About Time and Wavelength Dilation. No. 9182. EasyChair, 2022. Retrieved August 05, 2023, from https://easychair.org/publications/preprint/M7Zt 

[2]  Relativity : the Special and General Theory by Albert Einstein. (n.d.). Project Gutenberg. Retrieved October 28, 2022, from https://www.gutenberg.org/ebooks/5001